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A Division Algebra
Joseph Johnson

Hamilton had discovered what modern mathematicians term a skew field. That is, the system satisfies the following axioms applied to the direct sum of four real additive groups, under Hamilton's definitions of equality, addition, and multiplication. Let tex2html_wrap_inline164 be any quaternions with tex2html_wrap_inline166
(1) Closure of (a) addition and (b) multiplication. Both the addition and multiplication of two quaternions yields another quaternion.
(2) Commutativity of (a) addition and falsity of (b) multiplication commutativity. It is easy to see that tex2html_wrap_inline168 . Of course tex2html_wrap_inline170 in general.
(3) Associativity of (a) addition and (b) multiplication. Clearly tex2html_wrap_inline172 , by properties of real numbers. The proof of multiplication associativity has been described as ``torture" by at least one mathematician.
(4) (a) Addititive and (b) multiplicative identities. The addititve identity is simply 0.

tex2html_wrap_inline176

The mulitiplicative identity is 1=(1+0i+0j+0k).
(5) (a) Additive and (b) multiplicative inverses. The additive inverse of tex2html_wrap_inline180 is simply tex2html_wrap_inline182 . Finding the multiplicative inverse of tex2html_wrap_inline180 is more detailed. The use of the aforementioned modulus function, tex2html_wrap_inline134 , is crucial.

displaymath188

where tex2html_wrap_inline190 . This can be derived as a consequence of conjugacy in Quaternions (see Elements Paragraph 187). Every quaternion tex2html_wrap_inline180 has a conjugate tex2html_wrap_inline194 such that tex2html_wrap_inline196 is a real number. This allows us to set tex2html_wrap_inline198 . The explicit conjugate of tex2html_wrap_inline200 is tex2html_wrap_inline202 (which is analogous to conjugacy of complex numbers).
(6) Distributive laws. It seems almost amazing that the Quaternions are in fact distributive. That is

displaymath204

If this axiom did not hold, the Quaternions would be utterly uninteresting. It would represent two separate group structures (after deleting 0 in the multiplicative case) - hardly impressive to most mathematicians.

| The Abstractor | Elementary Notions | A Fourth Dimension | A Division Algebra
| Closure | An Example of Hamilton's Work | Bibliography | Back to the front page